Harmonic analysis on solvable Lie groups associated with constructions of induced representations

与诱导表示构造相关的可解李群的调和分析

基本信息

批准号：
15540171
负责人：
INOUE Junko
金额：
$ 1.41万
依托单位：
Tottori University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2003
资助国家：
日本
起止时间：
2003 至 2005
项目状态：
已结题

项目摘要

1.According to the orbit method, an irreducible unitary representation of an exponential solvable Lie group is realized as an induced representation from a unitary character of a connected subgroup on a space of L^2-functions of the homogeneous space diffeomorphic with R^n. When the group is nilpotent, the space of C^∞ vectors coincides with the space of Schwartz functions of R^n, and it fits in with a description of the Fourier transform for the group. However, when the group is general exponential, the space of C^∞ vectors does not have such simple descriptions. As a collaboration with J.Ludwig (University of Metz, France), we investigated C^∞ vectors for irreducible representations of exponential groups G, and obtained the following results : Taking a nilpotent ideal which includes the derived ideal of the Lie algebra of G, and choosing a real polarization adapted to the ideal, we realize the representation on a space of L^2-functions on the homogeneous space. We treat a subspace consisting of functions with some rapidly decreasing property associated with the nilpotent ideal, and show that it can be embedded as the space of C^∞ vectors for an irreducible representation of a group containing G. Using the space, we also describe a certain space of functions included in the image of Fourier transforms of L^1-functions on G of finite ranks. Furthermore, specific descriptions of the space of C^∞ vectors for some special classes of representations are obtained.2.I investigated generalized holomorphically induced representations, and described associated semiinvariant vectors for some examples of low dimensional solvable Lie groups. Their descriptions depend on particular group structures.

1.根据轨道方法，指数可解李群的不可约酉表示被实现为由与 R^n 齐次空间微分同胚的 L^2-函数空间上连通子群的酉性特征导出的表示当群幂零时，C^∞ 向量的空间与 R^n 的 Schwartz 函数的空间一致，并且符合的傅里叶变换的描述。然而，当群是一般指数群时，C^∞ 向量的空间没有如此简单的描述。作为与 J.Ludwig（法国梅斯大学）的合作，我们研究了 C^∞ 向量的不可约表示。指数群 G，并得到以下结果：取一个幂零理想，其中包括 G 的李代数的导出理想，并选择适合该理想的实极化，我们实现了空间上的表示我们处理齐次空间上的 L^2 函数，该子空间由具有与幂零理想相关的某些快速递减特性的函数组成，并证明它可以嵌入为 C^∞ 向量的空间，以实现群的不可约表示。利用该空间，我们还描述了有限秩G上的L^1函数的傅立叶变换图像中包含的特定函数空间。此外，还具体描述了某些C^∞向量的空间。 2.我研究了广义全纯诱导表示，并描述了低维可解李群的一些示例的相关半不变向量，它们的描述取决于特定的群结构。